Trees and treelike structures in dense digraphs

Abstract We prove that every oriented tree on n n $n$ vertices with bounded maximum degree appears as a spanning subdigraph of every directed graph on n n $n$ vertices with minimum semidegree at least n divided by 2 plus normal o left parenthesis n right parenthesis n / 2 + o ( n ) $n/2+{\mathrm{o}}(n)$ . This can be seen as a directed graph analogue of a well-known theorem of Komlós, Sárközy, and Szemerédi. Our result for trees follows from a more general result, allowing the embedding of arbitrary orientations of a much wider class of spanning ‘tree-like’ structures, such as collections of at most upper O left parenthesis n Superscript 0.99 Baseline right parenthesis O ( n 0.99 ) $O(n^{0.99})$ pairwise vertex-disjoint cycles and subdivisions of graphs upper H H $H$ with StartAbsoluteValue upper H EndAbsoluteValue less than exp left parenthesis StartRoot upper O left parenthesis log n right parenthesis EndRoot right parenthesis | H | < exp ⁡ ( O ( log ⁡ n ) ) $|H|\lt \exp \bigl (\sqrt {\textrm {O}(\log n)}\,\bigr )$ in which each edge is subdivided at least once.